To determine which of the given trinomials is a perfect square of a binomial, let's check each one step by step:
### 1. [tex]\( 16x^2 + 12x + 9 \)[/tex]
A perfect square trinomial takes the form:
[tex]\[ (ax + b)^2 = a^2x^2 + 2abx + b^2 \][/tex]
1. Write down the coefficients: [tex]\( a^2 = 16 \)[/tex], [tex]\( 2ab = 12 \)[/tex], [tex]\( b^2 = 9 \)[/tex].
2. Solve for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
- [tex]\( a^2 = 16 \)[/tex] implies [tex]\( a = 4 \)[/tex] or [tex]\( a = -4 \)[/tex].
- [tex]\( b^2 = 9 \)[/tex] implies [tex]\( b = 3 \)[/tex] or [tex]\( b = -3 \)[/tex].
3. Verify [tex]\( 2ab = 12 \)[/tex]:
- If [tex]\( a = 4 \)[/tex] and [tex]\( b = 3 \)[/tex], [tex]\( 2ab = 2 \cdot 4 \cdot 3 = 24 \)[/tex].
- If [tex]\( a = 4 \)[/tex] and [tex]\( b = -3 \)[/tex], [tex]\( 2ab = 2 \cdot 4 \cdot (-3) = -24 \)[/tex].
- If [tex]\( a = -4 \)[/tex] and [tex]\( b = 3 \)[/tex], [tex]\( 2ab = 2 \cdot (-4) \cdot 3 = -24 \)[/tex].
- If [tex]\( a = -4 \)[/tex] and [tex]\( b = -3 \)[/tex], [tex]\( 2ab = 2 \cdot (-4) \cdot (-3) = 24 \)[/tex].
None of these matches [tex]\( 12 \)[/tex].
### 2. [tex]\( 9x^2 - 9x + 1 \)[/tex]
1. Write down the coefficients: [tex]\( a^2 = 9 \)[/tex], [tex]\( 2ab = -9 \)[/tex], [tex]\( b^2 = 1 \)[/tex].
2. Solve for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
- [tex]\( a^2 = 9 \)[/tex] implies [tex]\( a = 3 \)[/tex] or [tex]\( a = -3 \)[/tex].
- [tex]\( b^2 = 1 \)[/tex] implies [tex]\( b = 1 \)[/tex] or [tex]\( b = -1 \)[/tex].
3. Verify [tex]\( 2ab = -9 \)[/tex]:
- If [tex]\( a = 3 \)[/tex] and [tex]\( b = 1 \)[/tex], [tex]\( 2ab = 2 \cdot 3 \cdot 1 = 6 \)[/tex] (no match).
- If [tex]\( a = 3 \)[/tex] and [tex]\( b = -1 \)[/tex], [tex]\( 2ab = 2 \cdot 3 \cdot (-1) = -6 \)[/tex] (no match).
- If [tex]\( a = -3 \)[/tex] and [tex]\( b = 1 \)[/tex], [tex]\( 2ab = 2 \cdot (-3) \cdot 1 = -6 \)[/tex] (no match).
- If [tex]\( a = -3 \)[/tex] and [tex]\( b = -1 \)[/tex], [tex]\( 2ab = 2 \cdot (-3) \cdot (-1) = 6 \)[/tex] (no match).
### 3. [tex]\( 16x^2 + 40x + 25 \)[/tex]
1. Write down the coefficients: [tex]\( a^2 = 16 \)[/tex], [tex]\( 2ab = 40 \)[/tex], [tex]\( b^2 = 25 \)[/tex].
2. Solve for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
- [tex]\( a^2 = 16 \)[/tex] implies [tex]\( a = 4 \)[/tex] or [tex]\( a = -4 \)[/tex].
- [tex]\( b^2 = 25 \)[/tex] implies [tex]\( b = 5 \)[/tex] or [tex]\( b = -5 \)[/tex].
3. Verify [tex]\( 2ab = 40 \)[/tex]:
- If [tex]\( a = 4 \)[/tex] and [tex]\( b = 5 \)[/tex], [tex]\( 2ab = 2 \cdot 4 \cdot 5 = 40 \)[/tex] (matches).
- If [tex]\( a = 4 \)[/tex] and [tex]\( b = -5 \)[/tex], [tex]\( 2ab = 2 \cdot 4 \cdot (-5) = -40 \)[/tex] (no match).
- If [tex]\( a = -4 \)[/tex] and [tex]\( b = 5 \)[/tex], [tex]\( 2ab = 2 \cdot (-4) \cdot 5 = -40 \)[/tex] (no match).
- If [tex]\( a = -4 \)[/tex] and [tex]\( b = -5 \)[/tex], [tex]\( 2ab = 2 \cdot (-4) \cdot (-5) = 40 \)[/tex] (matches).
So, [tex]\( 16x^2 + 40x + 25 \)[/tex] is a perfect square trinomial, and it matches the format [tex]\((4x + 5)^2\)[/tex] or [tex]\((-4x - 5)^2\)[/tex].
### 4. [tex]\( 25x^2 - 80x + 16 \)[/tex]
1. Write down the coefficients: [tex]\( a^2 = 25 \)[/tex], [tex]\( 2ab = -80 \)[/tex], [tex]\( b^2 = 16 \)[/tex].
2. Solve for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
- [tex]\( a^2 = 25 \)[/tex] implies [tex]\( a = 5 \)[/tex] or [tex]\( a = -5 \)[/tex].
- [tex]\( b^2 = 16 \)[/tex] implies [tex]\( b = 4 \)[/tex] or [tex]\( b = -4 \)[/tex].
3. Verify [tex]\( 2ab = -80 \)[/tex]:
- If [tex]\( a = 5 \)[/tex] and [tex]\( b = 4 \)[/tex], [tex]\( 2ab = 2 \cdot 5 \cdot 4 = 40 \)[/tex] (no match).
- If [tex]\( a = 5 \)[/tex] and [tex]\( b = -4 \)[/tex], [tex]\( 2ab = 2 \cdot 5 \cdot (-4) = -40 \)[/tex] (no match).
- If [tex]\( a = -5 \)[/tex] and [tex]\( b = 4 \)[/tex], [tex]\( 2ab = 2 \cdot (-5) \cdot 4 = -40 \)[/tex] (no match).
- If [tex]\( a = -5 \)[/tex] and [tex]\( b = -4 \)[/tex], [tex]\( 2ab = 2 \cdot (-5) \cdot (-4) = 40 \)[/tex] (no match).
### 5. [tex]\( -2x^2 + 4x - 1 \)[/tex]
1. Since the leading coefficient is negative, it's not a standard form for a perfect square trinomial.
Therefore, among the given options, the trinomial [tex]\( 16x^2 + 40x + 25 \)[/tex] is a perfect square of a binomial.
The correct answer is:
[tex]\[ \boxed{16x^2 + 40x + 25} \][/tex]
